An inverse mapping theorem for blow-Nash maps on singular spaces

TL;DR

This paper proves that under certain Jacobian conditions, a blow-Nash homeomorphism on singular real algebraic sets has an inverse that is also blow-Nash, using motivic integration and virtual Poincaré polynomial techniques.

Contribution

It extends the class of blow-Nash maps for which the inverse retains the blow-Nash property, generalizing previous results to maps that are generically one-to-one.

Findings

Inverse of blow-Nash homeomorphism is also blow-Nash under Jacobian bounds

Uses motivic integration and virtual Poincaré polynomial in the proof

Generalizes Denef-Loeser change of variables lemma

Abstract

A semialgebraic map $f:X\to Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e. semialgebraic and real analytic) by composing with finitely many blowings-up with non-singular centers. We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition. The proof relies on motivic integration arguments and on the virtual Poincar\'e polynomial of McCrory-Parusi\'nski and Fichou. In particular, we need to generalize Denef-Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational.